Training instabilities in deep networks - loss spikes, oscillatory convergence, and gradient pathologies - are empirically prevalent but lack a rigorous operator-theoretic explanation. We show that the linearized update operators for practically used optimizers are generically non-normal: for Adam, non-normality is controlled by the commutator [H, M] between the Hessian and the diagonal adaptive preconditioner, while for SGD with momentum it arises from the augmented state-space structure of the update map. Applying non-normal stability theory to these operators, we derive a conservative pseudospectral precursor bound in which κ(V) serves as an early-warning indicator of transient amplification even when the spectral radius remains below one, and we establish that exceptional points of the update operator appear as the κ(V) -> \infty limiting case of this framework. Numerical experiments on two-layer networks confirm that the spectral radius ρ(J) provides no separation between stable and unstable training phases while κ(V) separates them by approximately one order of magnitude, complementing the classical sharpness criterion with a continuous severity measure of non-normal amplification. These results establish non-Hermitian operator theory as a useful and underexplored framework for neural network optimization stability, offering a diagnostic language and proof-of-concept benchmark for understanding adaptive optimization stability.
Most explanations of training instability focus on \emph{learning-rate criticality}, typically characterized by the Edge of Stability, beyond which optimization becomes unstable. We argue that, in practical deep neural network training, there is an additional and often overlooked \emph{weight-norm criticality}. This criticality is induced by the interaction between normalization (which introduces scale-invariant components) and weight decay (which persistently shrinks parameter norms). As the weight decay coefficient increases, the norms of scale-invariant weights are progressively driven toward zero. Meanwhile, the sharpness of the loss landscape increases rapidly, destabilizing the optimization dynamics and resulting in abrupt loss spikes. This perspective provides a rationale for why weight penalties can improve generalization yet cannot be made arbitrarily strong: excessive decay drives scale-invariant weight norms past a critical boundary and destabilizes training. Our work provides a new mechanistic understanding of loss spikes through the lens of \emph{weight-norm criticality}. Moreover, \emph{weight-norm criticality} yields testable predictions that we validate empirically in networks with scale-invariant components, providing empirical support for the proposed mechanism.
Normalization renders large parts of neural networks effectively scale invariant, inducing a hidden feedback loop in which learning-rate schedules and weight decay interact through the parameter norm to control the effective step taken by the optimizer. We show that this interaction is governed by an exact discrete-time law: a single scalar quantity captures all schedule and decay forcing, while norm growth induces an opposing geometric self-quenching effect. This yields a sharp boundary that cleanly separates contraction- and expansion-dominated effective learning rate regimes. To understand the underlying mechanism, we provide exact analysis of a fully solved normalized regression model where the dynamics reduce to two dimensions and show that the balance point is intrinsically unstable, implying that constant learning rate with weight decay cannot stably maintain an interior equilibrium and instead produces recurrent behavior driven by discrete-time Jacobian structure. We further extend this perspective across optimizers through unified homogeneous-optimizer framework that reveals a structural dichotomy in self-quenching strength, providing a first-principles explanation for why adaptive methods exhibit systematically weaker stabilization under normalization. Across dynamical systems and neural networks (MLP, CNN, GPT2 / MNIST, CIFAR, wikiText, OpenWebText), the predicted law holds with high precision and enables direct control of training via the identified scalar, with performance peaking sharply at the predicted boundary. Together, these results isolate a single governing quantity for scale-invariant optimization, providing a precise and actionable lens on training dynamics, optimizer behavior, and schedule design in modern deep learning. Code is available in https://github.com/shasanamin/normalized-optimization-dynamics.
Coupled gradient descent - where the update of one parameter depends on another - arises naturally in bilevel optimization, two-time-scale stochastic approximation, and generative adversarial networks. When the coupled Jacobian is block-triangular, asymptotic stability is determined by the spectral radii of the diagonal blocks, yet transient amplification before convergence can be arbitrarily large due to non-normality. We develop a sharp pseudospectral theory for block-triangular Jacobians J = [[A, 0], [C, D]], proving Kreiss-constant bounds of the form K(J) <= 2/(1-γ) + ||C||/(4(1-γ)) when ρ(A), ρ(D) <= γ< 1 and A, D are symmetric, and establishing matching minimax lower bounds. We characterize the critical coupling threshold for spectral instability and extend the theory to nearly self-referential systems via a Neumann-series perturbation framework. As a consequence, we obtain a finite-horizon O(K(J)^2 log(1/δ)) iteration complexity bound. Framed as scaling laws for stochastic two-time-scale optimization, our results expose a non-asymptotic, instance-dependent regime of high-dimensional learning dynamics that is invisible to spectral-radius analysis. Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.